How to do texture painting on a character?
It sounds to me like you don't have your UV maps set up properly. Did you unwrap the model after shaping it? What you're describing is something I've had happen when Blender is still trying to use the UV map for the object I started with, when extensive modifications have made that useless. For example, paint appearing on two parts of the mesh at once suggests that you have portions of your UV map overlapping each other.

You have two options, I think: first, you could UV unwrap. To do that, go to the "UV Editing" tab with one of the pieces of your model selected, select the whole thing, press "U" and then click "Unwrap". It may result in a pretty bad map, so you may need to look up some tutorials on UV unwrapping. You'll also need to unwrap each piece separately, and it'll probably be easiest to use a separate image texture for each one.

The second option is easier but not as good for detail -- you could just vertex paint. For that, just switch from "Object Mode" to "Vertex Paint Mode"; I'd also recommend looking up a detailed tutorial on using Vertex Paint, though it's not terribly complicated.

Oh, and one more option: if all you want is a single color, all of this is overkill -- it would make much more sense to just change the material to one that uses the right base color.